Risk measure forecast and model have been developed in order to not only provide better forecast but also preserve its (empirical) property especially coherent property. Whilst the widely used risk measure of Value-at-Risk (VaR) has shown its performance and benefit in many applications, it is in fact not a coherent risk measure. Conditional VaR (CoVaR), defined as mean of losses beyond VaR, is one of alternative risk measures that satisfies coherent property. There have been several extensions of CoVaR such as Modified CoVaR (MCoVaR) and Copula CoVaR (CCoVaR). In this paper, we propose another risk measure, called Dependent CoVaR (DCoVaR), for a target loss that depends on another random loss, including model parameter treated as random loss. It is found that our DCoVaR provides better forecast than both MCoVaR and CCoVaR. Numerical simulation is carried out to illustrate the proposed DCoVaR. In addition, we do an empirical study of financial returns data to compute the DCoVaR forecast for heteroscedastic process of GARCH(1,1). The empirical results show that the Gumbel Copula describes the dependence structure of the returns quite nicely and the forecast of DCoVaR using Gumbel Copula is more accurate than that of using Clayton Copula. The DCoVaR is superior than MCoVaR, CCoVaR and CoVaR to comprehend the connection between bivariate losses and to help us exceedingly about how optimum to position our investments and elevate our financial risk protection. In other words, putting on the suggested risk measure will enable us to avoid non-essential extra capital allocation while not neglecting other risks associated with the target risk. Moreover, in actuarial context, DCoVaR can be applied to determine insurance premiums while reducing the risk of insurance company.
Dependent Tail Value-at-Risk, abbreviated as DTVaR, is a copula-based extension of Tail Value-at-Risk (TVaR). This risk measure is an expectation of a target loss once the loss and its associated loss are above their respective quantiles but bounded above by their respective larger quantiles. In this paper, we propose nonparametric estimators for DTVaR and establish their property of consistency. Moreover, we also propose the variability measure around this expected value truncated by the quantiles, called the Dependent Conditional Tail Variance (DCTV). We use this measure for constructing confidence intervals of the DTVaR. Both parametric and nonparametric approaches for DTVaR estimations are explored. Furthermore, we assess the performance of DTVaR estimations using a proposed backtest based on the DCTV. As for the numerical study, we take an application in the insurance claim amount.
In this paper, we study a novel risk measure, which is a copula-based extension of tail valueat-risk (TVaR). This measure is called dependent tail value-at-risk (DTVaR), which is a generalization of TVaR. Moreover, we describe a second conditional tail moment of the tail distribution with the center being the DTVaR itself, which is called the dependent conditional tail variance (DCTV). Both DTVaR and DCTV contain two contraction parameters, which make them much more flexible than some of the more familiar measures of risk, such as TVaR and conditional tail variance (CTV). We derive analytical formulas of the DTVaR and DCTV for exponential risk associated with another risk where their dependence structure is represented by Farlie-Gumbel-Morgenstern (FGM) copula. This paper proposes an optimization method for DTVaR by applying two metaheuristic algorithms: spiral optimization (SpO) and particle swarm optimization (PSO). Furthermore, we perform SpO and PSO by utilizing DCTV and CTV to estimate two contraction parameters that maximize DTVaR. This work presents an application of DTVaR optimization in predicting the DTVaR of energy risk of New York Harbor (NYH) gasoline associated with energy risk of West Texas Intermediate (WTI) crude oil. We find that the values of the objective function using both algorithms converge to zero, which implies that the SpO and PSO algorithms are very suitable for application to DTVaR optimization. However, according to the values of the objective function, we find that the PSO algorithm is more suitable than the SpO algorithm in optimizing DTVaR.INDEX TERMS dependent tail value-at-risk, dependent conditional tail variance, FGM copula, metaheuristic algorithms, energy risk.
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